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Properties

Label 2-671-61.34-c1-0-16
Degree $2$
Conductor $671$
Sign $0.354 - 0.935i$
Analytic cond. $5.35796$
Root an. cond. $2.31472$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.729 − 0.529i)2-s + (−0.384 + 0.279i)3-s + (−0.366 + 1.12i)4-s + (0.465 − 1.43i)5-s + (−0.132 + 0.407i)6-s + (2.67 + 1.94i)7-s + (0.887 + 2.73i)8-s + (−0.857 + 2.63i)9-s + (−0.419 − 1.29i)10-s − 11-s + (−0.174 − 0.537i)12-s − 5.28·13-s + 2.98·14-s + (0.221 + 0.681i)15-s + (0.173 + 0.126i)16-s + (−0.665 + 2.04i)17-s + ⋯
L(s)  = 1  + (0.515 − 0.374i)2-s + (−0.222 + 0.161i)3-s + (−0.183 + 0.564i)4-s + (0.208 − 0.641i)5-s + (−0.0540 + 0.166i)6-s + (1.01 + 0.735i)7-s + (0.313 + 0.966i)8-s + (−0.285 + 0.879i)9-s + (−0.132 − 0.408i)10-s − 0.301·11-s + (−0.0503 − 0.155i)12-s − 1.46·13-s + 0.797·14-s + (0.0572 + 0.176i)15-s + (0.0434 + 0.0316i)16-s + (−0.161 + 0.497i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.354 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.354 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $0.354 - 0.935i$
Analytic conductor: \(5.35796\)
Root analytic conductor: \(2.31472\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 671,\ (\ :1/2),\ 0.354 - 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38209 + 0.954630i\)
\(L(\frac12)\) \(\approx\) \(1.38209 + 0.954630i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 + T \)
61 \( 1 + (-6.88 + 3.69i)T \)
good2 \( 1 + (-0.729 + 0.529i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.384 - 0.279i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.465 + 1.43i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-2.67 - 1.94i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + 5.28T + 13T^{2} \)
17 \( 1 + (0.665 - 2.04i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.23 + 1.62i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (1.35 - 4.18i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 - 5.76T + 29T^{2} \)
31 \( 1 + (2.52 + 1.83i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.31 - 1.68i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-4.04 - 2.93i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-2.13 - 6.58i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 - 1.75T + 47T^{2} \)
53 \( 1 + (-1.73 - 5.33i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.65 + 1.20i)T + (18.2 - 56.1i)T^{2} \)
67 \( 1 + (-4.03 + 12.4i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-1.10 - 3.38i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (2.55 + 7.86i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (4.00 + 12.3i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.48 - 1.80i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (2.82 - 2.05i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (13.1 + 9.55i)T + (29.9 + 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.01919920474642393157693569230, −9.821624472496703366374940270037, −8.860645923626592519379234025651, −8.058139885232523670733419542685, −7.48213125667574531440699079572, −5.68272900359444984780976190280, −4.95173942878175921320100484322, −4.55894920836200896179764476415, −2.86453128138533824618653481456, −1.95960317362954572017251871113, 0.795924772421287648473486586529, 2.51971593649760498494356030021, 4.04963803527937178173304281209, 4.93988322654121478019208087308, 5.75126077483268428591000364836, 6.89946014405339314147731248999, 7.24235617932645594869558988871, 8.548067459612922903733161066663, 9.748503120133876019285686932809, 10.30348598433961984062891932909

Graph of the $Z$-function along the critical line